Addresses the two class discrimination problem in the case that each class can be viewed as being composed of a finite number of types. The prior probabilities for the types comprising both classes are unknown, the class costs are known and the conditional densities for the types are real analytic functions. An algorithm is presented that can be used to estimate an optimal discriminant function that is robust in the minimax sense. The algorithm involves a search over a set of prior weights. The convergence properties of the algorithm are examined in a series of tests involving Gaussian mixture densities.>

A maximum likelihood approach is developed for a pattern recognition problem where the patterns are described by configurations of simple easily recognized parts called primitives. The approach is capable of dealing with three types of noise: measurement noise in the location and shape of observed primitives, undetected or missing primitives (leakage), and the unexpected appearance of extra primitives (false alarms). The approach is called combinatorial because the likelihood function dictates that observed primitives must be assigned to known primitives in all possible combinations. Due to the complexity of the likelihood function, practical classifiers must be based on likelihood function approximations. Several are proposed, and most of these are simple enough to be used in a gradient search strategy for recognizing distorted patterns with random orientations. Examples are included to show the characteristics of combinatorial classifier performance.

The properties of the ``magnifying glass'' method of clustering are discussed. These properties, which include unbiased and consistent estimation of the mean for Gaussian distributions and biased and inconsistent estimation of the mean for non-Gaussian distributions, lead to the development of a technique for testing data to determine whether or not it is Gaussian. An example of a non-Gaussian distribution is given to show the sensitivity of the proposed Gaussian test.

By proper design of a nearest-neighbor (NN) rule, it is possible to reduce effects of sample size in NN risk estimation. The 2-NN rule for the two-class problem eliminates the first-order effects of sample size. Since its asymptotic value is exactly half that of the 1-NN rule, it is possible to substitute the 2-NN rule for the 1-NN rule with a resultant increase in accuracy. For further stabilization of the risk estimate with respect to sample size, 2-NN polarization is suggested. Examples are included. The 2-NN approach is extended to M-class and 2k-NN.

A quadratic metric dAO (X, Y) =[(X - Y)T AO(X - Y)]¿ is proposed which minimizes the mean-squared error between the nearest neighbor asymptotic risk and the finite sample risk. Under linearity assumptions, a heuristic argument is given which indicates that this metric produces lower mean-squared error than the Euclidean metric. A nonparametric estimate of Ao is developed. If samples appear to come from a Gaussian mixture, an alternative, parametrically directed distance measure is suggested for nearness decisions within a limited region of space. Examples of some two-class Gaussian mixture distributions are included.

Classification error is analyzed for a situation where the number of possible classes may be on the order of a hundred or more. The error associated with classifying to a single class is shown to depend mainly on average nearest-neighbor distance between class means, noise level, and effective dimensionality of the class mean distribution and not much on other aspects of the distribution, noise correlation, or number of classes. Since single class error is large, separation of classes into groups is also explored. Group classification error has the same properties as single class error but the size of the error is moderated by the Bayes overlap between groups. Standard curves are provided to predict single class and group error. Also discussed are the effect of pattern blurring on classification error and the nearest-neighbor distance statistics throughout a distribution.

Given a general n-dimensional bimodal Gaussian mixture, this paper shows how unknown parameters may be found by the method of moments. Three cases are considered-equal modal probabilities, known but not necessarily equal probabilities, and all parameters unknown. The solution involves sample moments no higher than fourth order. For Gaussian mixtures where the number of modes is unknown, fourth-order moments can be used to count them, provided all modes have the same covariance matrix, and their multiplicity is not greater than data dimensionality. Examples of mode-counting and the determination of bimodal parameters are included.